Abstract- In this page, we find that the b-chromatic number on corona graph of subdivision-vertex path with path. Then corona graph of any graph with path, cycle and complete graph and cycle with path.

1. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 44-47
ISSN: 2278-2397
44
b-Chromatic Number of Subdivision Edge and
Vertex Corona
T.Pathinathan1
, A.Arokia Mary2
, D.Bhuvaneswari2
1
Department of Mathematics,Loyola college,Chennai
2
Department of Mathematics, St.Joseph’sCollege, Cuddalore, India
Email:arokia68@gmail.com,bhuvi64@gmail.com
Abstract - In this paper, we find that the b-chromatic number
on corona graph of subdivision-vertex path with path. Then
corona graph of any graph with path, cycle and complete graph
and cycle with path.
Keywords: b-chromatic number, corona graph, subdivision-
edge corona, subdivision-vertex corona, edge corona.
I. INTRODUCTION
The b-chromatic number of a graph , denoted by , is
the maximal integer k such that may have a b-coloring with
k colors. This parameter has been defined by Irving and
Manlove. The subdivision graph of a graph is the
graph obtained by inserting a new vertex into every edge of ,
we denote the set of such new vertices by .
In two new graph operations based on subdivision graphs,
subdivision-vertex join and subdivision-edge join. The corona
of two graphs was first introduced by R.Frucht and F.Harary in
[11]. And another variant of the corona operation, the
neighbourhood corona, was introduced in [12].
II. PRELIMINARIES
DEFINITION2.1.Let and be two graphs. Let
and take k copies of . The
corona is the graph obtained by joining each to
every vertex of the copy of , .
G2 G1
G1 ∘ G2
DEFINITION2.2. Let and be two graphs on disjoint
sets of and vertices, and edges respectively.
The edge corona 𝐺1 ⋄ 𝐺2 of and is defined as the
graph obtained by taking one copy of and copies of
and then joining two end-vertices of the edge of to
every vertex in the copy of .
Example: Let be the cycle of order 4 and be the
complete graph of order 2. The two edge coronas𝐺1 ⋄ 𝐺2.
G1 ⋄ G2
G (G)
b

G
( )
S G G
G
( )
I G
1
G 2
G
 
1 1 2
V(G ) , ,..., k
v v v
 2
G
1 2
G G Vi
th
i 2
G 1 i k
 
1
G 2
G
1
n 2
n 1
m 2
m
1
G 2
G
1
G 1
m
2
G th
i 1
G
th
i 2
G
1
G 2
G
2
k
2
G 1
G

2. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 44-47
ISSN: 2278-2397
45
III. SUBDIVISION-VERTEX AND SUBDIVISION-
EDGE CORONA
DEFINITION 3.1. The Subdivision-vertex Corona of two
vertex-disjoint graphs and , denoted by𝐺1 ⊙ 𝐺2 is the
graph obtained from and copies of , all
vertex-disjoint by joining the vertex of to every
vertex in the copy of .
DEFINITION3.2.The Subdivision-edge Corona of two vertex
disjoint graphs and denoted by𝐺1 ⊝ 𝐺2 is the graph
obtained from and copies of , all vertex-
disjoint, by joining the vertex of to every vertex in
the copy of .
Let is a graph on vertices and edges and is a
graph on vertices and edges then the subdivision-
vertex Corona 𝐺1 ⊙ 𝐺2 has vertices and
edges, and the Subdivision-edge Corona
𝐺1 ⊝ 𝐺2 has vertices and
edges.
THEOREM 3.3. Let 𝑃𝑛 be a path of 𝑛 vertices and 𝑃𝑚 be a path
of 𝑚 vertices. Then
𝜒𝑏(𝑃𝑛 ⊙ 𝑃𝑚) = {
2𝑛 − 1 𝑛 ≥ 𝑚
𝑚 + 1 𝑛 < 𝑚
.
PROOF.
Let 𝑉(𝑃𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛} and 𝑉(𝑃𝑚) = {𝑢1, 𝑢2, … , 𝑢𝑚}. let
𝑉(𝑃𝑛 ∘ 𝑃𝑚) = {𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢𝑖𝑗 ∶ 1 ≤ 𝑖 ≤ 𝑚; 1 ≤ 𝑗 ≤
𝑚}.By the definition of corona graph, each vertex of 𝑃𝑛 is
adjacent to every vertex of a copy of 𝑃𝑚.
Assign the following n-coloring for 𝑃𝑛 ⊙ 𝑃𝑚 as b-chromatic.
 For 1 ≤ 𝑖 ≤ 𝑛, assign the color 𝑐𝑖 to 𝑣𝑖.
 For 1 ≤ 𝑖 ≤ 𝑛, assign the color 𝑐𝑖 to 𝑢1𝑖, ∀ 𝑖 ≠ 1.
 For 1 ≤ 𝑖 ≤ 𝑛, assign the color 𝑐𝑖 to 𝑢2𝑖, ∀ 𝑖 ≠ 2.
 For 1 ≤ 𝑖 ≤ 𝑛, assign the color 𝑐𝑖 to 𝑢3𝑖, ∀ 𝑖 ≠ 3.
 For 1 ≤ 𝑖 ≤ 𝑛, assign the color 𝑐𝑖 to 𝑢𝑛𝑖, ∀ 𝑖 ≠ 𝑛.
 For 1 ≤ 𝑖 ≤ 𝑛, assign to vertex 𝑢𝑖𝑖 one of allowed colors.
Define 𝜒𝑏(𝑃3 ⊙ 𝑃2) = 5.
 For 1 ≤ 𝑖 ≤ 5, assign the color 𝑐𝑖 to 𝑣𝑖.
 For 1 ≤ ℓ ≤ 5, assign the color 𝑐𝑖 to 𝑢1𝑖∀ 𝑖 ≠ 1.
 For 1 ≤ ℓ ≤ 5, assign the color 𝑐𝑖 to 𝑢2𝑖∀ 𝑖 ≠ 2.
 For 1 ≤ ℓ ≤ 5, assign the color 𝑐𝑖 to 𝑢3𝑖∀ 𝑖 ≠ 3.
𝜒𝑏(P3 ⊙ P2) = 5
IV.VERTEX CORONA
4.1 GRAPHS WITH PATH
THEOREM4.1.1 Let Gbe a simple graph on vertices. Then
PROOF:
Let and .
Let .
By the definition of Corona graph each vertex of G is adjacent
to every vertex of copy of . i.e) every vertex
is adjacent to every vertex from the set
.
Assign the following -coloring for as b-chromatic
 For , assign the color to .
 For , assign the color to , .
 For , assign the color to , .
 For , assign the color to , .
 For , assign the vertex one of allowed
colors-such color exists because and
.
Let us assume that is greater than ie)
there must be at least
vertices of degree in all with distinct colors, and
each adjacent to vertices of all of the other colors. But then
these must be the vertices . Since these are only
1
G 2
G
1
( )
S G 1
V(G ) 2
G
th
i 1
V(G )
th
i 2
G
1
G 2
G
1
( )
S G 1
I(G ) 2
G
th
i 1
I(G )
th
i 2
G
1
G 1
n 1
m 2
G
2
n 2
m
1 2 1
(1 )
n n m
 
1 1 2 2
2 ( )
m n n m
 
1 2 1
(1 )
m n n
  1 2 2
(2 )
m n m
 
n
 1,
(G P ) n
b n n
 
 for
for
3
3
n
n


 
1 2
V(G) , ,..., n
v v v
  
1 2
V(P ) , ,...,
n n
u u u

   
V(G ) :1 :1 ;1
n i ij
P v i n U i n j n
       
n Pn
V( )
i
v G

 
:1
ij
u j n
 
n G n
P
1 i n
  i
c i
v
1 i n
  i
c 1i
u 1.
i
 
1 i n
  i
c 2i
u 2.
i
 
1 i n
  i
v ni
u .
i n
 
1 i n
  ii
u
2 deg(u ) 3
ii
 
3
n 
(G P )
b n
 n
(G P ) 1
b n n
   3
n
  1
n
n G Pn
 
1 2
, ,..., n
v v v
1
3
2
1 2 4 5
3
1 2
2

3. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 44-47
ISSN: 2278-2397
46
ones with degree at least . This is a contradiction. b-coloring
with colors is impossible.
Thus we have . Hence ,
.
Define b-coloring of with 5 colors in the
following way:
 For , assign the color to .
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
We have
. Hence .
χb(G ∘ P4) = 5
4.2. GRAPH WITH CYCLE
THEOREM4.2.1. Let be a simple graph on vertices
. Then .
Proof:
Let and .
Let . By the
definition of Corona graph, each vertex of G is adjacent to
every vertex of a copy of . i.e., every vertex is
adjacent to every vertex from the set .
Assign the following -coloring for as b-chromatic:
 For , assign the color to .
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
 For , assign to vertex one of allowed
colors-such color exists, because and .
. Assume that is greater than
, ie) , there must be at least
vertices of degree in , all with distinct
colors, and each adjacent to vertices of all of the other colors.
But then these must be the vertices . Since these
are only ones with degree at least . This is the contradiction
b-coloring with color is impossible.
We have . Hence ,
.
𝜒𝑏(𝐺 ∘ 𝐶3) = 4.
𝜒𝑏(𝐺 ∘ 𝐶3) = 4.
4.3. CYCLE WITH PATH
THEOREM 4.3.1Let be a cycle of vertices and be a
path of vertices. Then
Proof:
Let and
.Let , By the
definition of Corona graph, each vertex of G is adjacent to
every vertex of a copy of . i.e., every vertex is
adjacent to every vertex from the set .
n
1
n
(G P )
b n n
  (G P )
b n n
 
3
n
 
4
G P ( V(G) ) 4

1 5
i
  i
c i
v
1 5
l
  l
c 1l
u 1.
l
 
1 5
l
  l
c 2l
u 2.
l
 
1 5
l
  l
c 3l
u 3.
l
 
1 5
l
  l
c 4l
u 4.
l
 
1 5
l
  l
c 5l
u 5.
l
 
4
(G P ) 5
b
  (G P )
b n n
  3
n
 
G n
3
n  (G C )
b n n
 
 
1 2
V(G) , ,..., n
v v v
  
1 2
V(C ) , ,...,
n n
u u u

   
V(G ) :1 :1 ;1
n i ij
C v i n u i n j n
       
n
C V( )
i
v G

 
:1
ij
u j n
 
n G n
C
1 i n
  i
c i
v
1 i n
  i
c 1i
u 1.
i
 
1 i n
  i
c 2i
u 2.
i
 
1 i n
  i
c ni
u .
i n
 
1 i n
  ii
u
deg(u ) 3
ii  3
n 
(G C )
b n n
  (G C )
b n

n (G C ) 1
b n n
   3
n
 
1
n n G Cn
 
1 2
, ,..., n
v v v
n
1
n
(G C )
b n n
  (G C )
b n n
 
3
n
 
n
C n Pn
n (C P )
b n n n
 
 
1 2
V(C ) , ,...,
n n
v v v
  
1 2
V(P ) , ,...,
n n
u u u

   
V(C ) :1 :1 ;1
n n i ij
P v i n u i n j n
       
n
P V(C )
i n
v 
 
:1
ij
u j n
 
5 1
1
5
4
2 3
2
3
4
5
5
4
3
1 2
4
4
1
2
5
3
3
2
2
2
2
1
1
2
3
4
2
1
3
4
1
3
2
4

4. Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 03, Issue: 01 June 2014, Pages: 44-47
ISSN: 2278-2397
47
. Assign the following b-coloring for
.
 For , assign the color to .
 For , assign the color to ,
 For , assign the color to ,
…………………………………
………………………………….
………………………………….
……………………………………
 For , assign the color to ,
 For , assign to vertex one of allowed colors
such color exists, because and .
We have . Hence .
We have .
χb(C3 ∘ P2) = 3.
4.4. GRAPH WITH COMPLETE GRAPH
Theorem4.4.1 :Let G be a simple graph on vertices. Then
Proof.
Let and .
Let𝑉(𝐺 ∘ 𝐾𝑛) = {𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢𝑖𝑗: 1 ≤ 𝑖 ≤ 𝑛; 1 ≤ 𝑗 ≤ 𝑛}.
By the definition of corona graph, each vertex of G is adjacent
to every vertex of a copy of . i.e., every vertex
is adjacent to every vertex from the set
.
Assign the following -coloring for as b-
chromatic:
 For , assign the color to .
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
 For , assign the color to ,
……………………………………
……………………………………
……………………………………
...………………………………….
 For , assign the color to ,
 For , assign the color and Therefore,
Let us assume that is greater than , i.e.,
, there must be at least vertices
of degree in , all with distinct colors, and each
adjacent to vertices of all of the other colors. But then these
must be the vertices , since these are only ones with
degree at least . This is the contradiction, b-coloring with
colors is impossible. Thus, we have
. Hence,.
V. CONCLUSION
In this existing subdivision-vertex corona graphs, they used
spectrum. Here we study about subdivision- vertex corona
graph using b-chromatic number.
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(C P )
b n n n
 
n n
C P
1 i n
  i
c i
v
1 i n
  i
c 1i
u 1.
i
 
1 i n
  i
c 2i
u 2.
i
 
1 i n
  i
c ni
u .
i n
 
1 i n
  ii
u
2 deg(u ) 3
ii
  3
n 
(C P )
b n n n
  (C P )
b n n n
 
3 2
(C P ) 3
b
 
n
  .
1

 n
K
G n
b 

 
1 2
V(G) , ,..., n
v v v
  
1 2
V(K ) , ,...,
n n
u u u

n
K
V( )
i
v G

 
:1
ij
u j n
 
1
n G n
K
1 i n
  i
c i
v
1 l n
  l
c 1l
u 1.
l
 
1 l n
  l
c 2l
u 2.
l
 
1 l n
  l
c 3l
u 3.
l
 
1 l n
  l
c 4l
u 4.
l
 
1 l n
  l
c nl
u .
l n
 
1 l n
  1
n
c  ll
u
  .
1

 n
K
G n
b 

 
n
b K
G 
 1
n
  2

 n
K
G n
b 
 2
n
1
n G n
K
1 2
, ,... n
v v v
1
n
2
n
  .
1

 n
K
G n
b 

4
P
1
2
3
3
1
2 1
2
3